Skin Effects

It is difficult for electromagnetic waves to penetrate into a conductor because of severe impedance matching. The impedance of a conductor is given by

where sigma (Siemens/m) is the conductivity. Noting
we may write
Even in microwave frequency regime, this is much smaller than the impedance of free space (377 Ohms) because of large value of conductivity of ordinary conductors. The imaginary part of the impedance indicates strong dissipation of electromagnetic energy into heat.

Electromagnetic waves can penetrate into a conductor to a depth of order

This quantity is called the "skin depth." When a dc field is suddenly applied to a conductor, the filed will eventually fully penetrate but it takes about
for the field to do so. Here, a is a characteristic size of a conductor (e.g., thickness of slab, radius of sphere, etc.) This is called "skin time."

Animation at the top shows penetration of an oscillating field, and bottom animation shows transient penetration of a dc field into a slab.

with(plots):
animate(cos(2*Pi*.1*t)*Heaviside(-x)+Heaviside(x)*exp(-x)*cos(x-2*Pi*.1*t),x=-2..7,t=0..9,frames=20,color=red,numpoints=50);

with(plots):
animate(Heaviside(abs(x)-.5)+(1.-(4./Pi)*(cos(Pi*x)*exp(-t)-cos(3*Pi*x)*exp(-9*t)/3+cos(5*Pi*x)*exp(-25*t)/5-cos(7*Pi*x)*exp(-49*t)/7+cos(9*Pi*x)*exp(-81*t)/9-cos(11*Pi*x)*exp(-121*t)/11+cos(13*Pi*x)*exp(-169*t)/13))*Heaviside(.5-abs(x)),x=-2..2,t=0.01..10,frames=50,color=red,numpoints=100);